2.7. Energy integral

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2.7. Energy integral

  1. Energy integral: wave equation
  2. Energy integral: Maxwell equation
  3. Energy integral: Elasticity equations

Energy integral: wave equation

Consider multidimensional wave equation \begin{equation} u_{tt}-c^2 \Delta u=0. \label{eq-2.7.1} \end{equation} Recall that $\Delta =\nabla\cdot \nabla=\partial_x^2+\partial_y^2+\partial_z^2$ (the number of terms depends on dimension). Multiplying by $u_t$ we get in the left-hand expression \begin{align*} u_t u_{tt}-c^2 u_t\nabla^2 u= &\partial_t \bigl(\frac{1}{2}u_t^2 \bigr)+ \nabla \cdot (-c^2 u_t \nabla u )+ c^2 \nabla u_t \cdot \nabla u \\ =&\partial_t \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u_t \nabla u \bigr). \end{align*} So we arrive to \begin{equation} \partial_t \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)+ \nabla \cdot \bigl(-c^2 u_t \nabla u \bigr). \label{eq-2.7.2} \end{equation} This is energy conservation law in the differential form. Here \begin{equation} e= \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr) \label{eq-2.7.3} \end{equation} is a density of energy and \begin{equation} \mathbf{S}=-c^2 u_t \nabla u \label{eq-2.7.4} \end{equation} is a vector of energy flow.

Then if we fix a volume (or an area in $2D$ case, or just an interval in $1D$ case) $V$ and introduce a full energy in $V$ at moment $t$ \begin{equation} E_V(t)= \iiint _V \bigl(\frac{1}{2}u_t^2 + \frac{1}{2}c^2 |\nabla u|^2\bigr)\, dV \label{eq-2.7.5} \end{equation} then \begin{equation} E_V(t_2) - E_V(t_1) + \int_{t_1}^{t_2}dt \iint_\Sigma \mathbf{S}\cdot \mathbf{n}\,d\sigma =0 \label{eq-2.7.6} \end{equation} where $\Sigma$ is the surface bounding $V$, $d\sigma$ is an element of the surface area, and $\mathbf{n}$ is an unit exterior normal to $\Sigma$.

Energy integral: Maxwell equation

Similarly, Maxwell equations without charges and currents are \begin{align} &\varepsilon \mathbf{E}_t = c\nabla \times \mathbf{H},\label{eq-2.7.7}\\ &\mu\mathbf{H}_t = -c\nabla \times \mathbf{E},\label{eq-2.7.8}\\ &\nabla \cdot \varepsilon\mathbf{E}=\nabla \cdot \mu\mathbf{H}=0. \label{eq-2.7.9} \end{align} Here $\mathbf{E}, \mathbf{H}$ are intensities of electric and magnetic field respectively, $c$ is the speed of light in the vacuum, $\varepsilon$ and $\mu$ are dialectric and magnetic characteristics of the media ($\varepsilon\ge 1$, $\mu \ge 1$ and $\varepsilon=\mu=1$ in the vacuum).

Multiplying (taking an inner product) (\ref{eq-2.7.7}) by $\mathbf{E}$ and (\ref{eq-2.7.8}) by $\mathbf{H}$ and adding we arrive to \begin{equation*} \partial_t \bigl(\frac{1}{2}\varepsilon|\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)= c\bigl(\mathbf{E}\cdot (\nabla \times \mathbf{H}) - \mathbf{H}\cdot (\nabla \times \mathbf{E})\bigr)= -c\nabla \cdot \bigl(\mathbf{E}\times \mathbf{H}\bigr) \end{equation*} where the last equality follows from vector calculus.

Then \begin{equation} \partial_t \bigl(\frac{1}{2}\varepsilon |\mathbf{E}|^2 + \frac{1}{2}\mu |\mathbf{H}|^2\bigr)+ \nabla \cdot \bigl(c\mathbf{E}\times \mathbf{H}\bigr)=0. \label{eq-2.7.10} \end{equation} In the theory of electromagnetism \begin{equation} e=\frac{1}{2}\bigl(\varepsilon |\mathbf{E}|^2 + \mu |\mathbf{H}|^2\bigr) \label{eq-2.7.11} \end{equation} is again density of energy and \begin{equation} \mathbf{S}=c \mathbf{E}\times \mathbf{H} \label{eq-2.7.12} \end{equation} is a vector of energy flow (aka Poynting vector).

Remark 1. $\frac{c}{\sqrt{\mu \varepsilon}}$ is the speed of light in the given media.

Remark 2. In inhomogeneous media $\varepsilon$ and $\mu$ depend on $(x,y,z)$; in anisotropic media (crystals) $\varepsilon$ and $\mu$ are matrices and then $e=\frac{1}{2}\bigl(\varepsilon \mathbf{E} \cdot \mathbf{E} + \mu \mathbf{H}\cdot \mathbf{E}\bigr)$.

Elasticity equations

Elasticity equations in homogeneous isotropic media are \begin{equation} \mathbf{u}_{tt}= 2\mu \Delta \mathbf{u} + \lambda \nabla (\nabla \cdot \mathbf{u}) \label{eq-2.7.13} \end{equation} where $\mathbf{u}$ is a displacement and $\lambda>0$, $\mu>0$ are Lamé parameters.

Problem 1. Multiplying (taking an inner product) (\ref{eq-2.7.13}) by $\mathbf{u}_t$ write conservation law in the differential form. What are $e$ and $\mathbf{S}$?

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